Multiple elliptic integrals and differential equations

Practical problems encountered within physics and engineering, such as problems concerning applications of three-dimensional lattices, often require the evaluation of definite integrals of the form ∫αβF(u)K(G(u)) dufor the complete elliptic integral K of the first kind and for algebraic functions F and G. Building on each of the authors' past techniques on the evaluation of and application of multiple elliptic integrals, we present techniques for evaluating integrals of the above form, using an ODE of Picard–Fuchs type and a second-order PDE. Specifically, we obtain new expressions of Ramanujan-type series of level 4 converted into integrals of the above form. This is achieved through new applications of the associated elliptic integral singular values and together with a Fourier–Legendre expansion generalizing the Bauer–Ramanujan identity.